## A quantitative ergodic theory proof of Szemerédi’s theorem (2004)

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@MISC{Tao04aquantitative,

author = {Terence Tao},

title = {A quantitative ergodic theory proof of Szemerédi’s theorem},

year = {2004}

}

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### Abstract

A famous theorem of Szemerédi asserts that given any density 0 < δ ≤ 1 and any integer k ≥ 3, any set of integers with density δ will contain infinitely many proper arithmetic progressions of length k. For general k there are essentially four known proofs of this fact; Szemerédi’s original combinatorial proof using the Szemerédi regularity lemma and van der Waerden’s theorem, Furstenberg’s proof using ergodic theory, Gowers’ proof using Fourier analysis and the inverse theory of additive combinatorics, and Gowers’ more recent proof using a hypergraph regularity lemma. Of these four, the ergodic theory proof is arguably the shortest, but also the least elementary, requiring in particular the use of transfinite induction (and thus the axiom of choice), decomposing a general ergodic system as the weakly mixing extension of a transfinite tower of compact extensions. Here we present a quantitative, self-contained version of this ergodic theory proof, and which is “elementary ” in the sense that it does not require the axiom of choice, the use of infinite sets or measures, or the use of the Fourier transform or inverse theorems from additive combinatorics. It also gives explicit (but extremely poor) quantitative bounds.

### Citations

250 |
Recurrence in Ergodic Theory and Combinatorial Number Theory
- Furstenberg
- 1981
(Show Context)
Citation Context ...compact” operator uniformly in choice of h and (gh)h∈H. 11 This observation was motivated by the use of relatively almost periodic functions in the ergodic theory arguments of Furstenberg [10], [15], =-=[11]-=- and later authors. the electronic journal of combinatorics 13 (2006), #R99 17 (15)sWe first observe that the construction A ↦→ UAP [A] maps shift-invariant Banach algebras to shift-invariant Banach a... |

249 | On sets of integers containing no k elements in arithmetic progressions
- Szemerédi
(Show Context)
Citation Context ...ombinatorics 13 (2006), #R99 1sSee for instance [22] for the standard “colour focusing” proof; another proof can be found in [36]. This theorem was then generalized substantially in 1975 by Szemerédi =-=[39]-=- (building upon earlier work in [33], [38]), answering a question of Erdős and Turán [8], as follows: Theorem 1.2 (Szemerédi’s theorem). For any integer k ≥ 1 and real number 0 < δ ≤ 1, there exists a... |

214 | Szemerédi’s Regularity Lemma and its applications in graph theory - Komlós, Simonovits - 1996 |

164 | The primes contain arbitrarily long arithmetic progressions
- Green, Tao
(Show Context)
Citation Context ...combinatorics, it has spurred much further research in other areas such as graph theory, ergodic theory, Fourier analysis, and number theory; for instance it was a key ingredient in the recent result =-=[23]-=- that the primes contain arbitrarily long arithmetic progressions. Despite the variety of proofs now available for this theorem, however, it is still regarded as a very difficult result, except when k... |

155 |
Ergodic behavior of diagonal measures and a theorem of Szemerédi on arithmetic progressions
- Furstenberg
- 1977
(Show Context)
Citation Context ...of Ackermann type), due mainly to the reliance on the van der Waerden theorem and the regularity lemma, both of which have notoriously bad dependence of the constants. Shortly afterwards, Furstenberg =-=[10]-=- (see also [15], [11]) introduced what appeared to be a completely different argument, transferring the problem into one of recurrence in ergodic theory, and solving that problem by a number of ergodi... |

152 | A new proof of Szemerédi’s theorem
- Gowers
(Show Context)
Citation Context ... of randomness and structure in the function f. We shall do this by means of two families of norms 9 : the Gowers uniformity norms �f� U 0 ≤ �f� U 1 ≤ . . . ≤ �f� U k−1 ≤ . . . ≤ �f�L ∞ introduced in =-=[20]-=- (and studied further in [26], [23]) and a new family of norms, the uniform almost periodicity norms �f� UAP 0 ≥ �f� UAP 1 ≥ . . . ≥ �f� UAP k−2 ≥ . . . ≥ �f�L ∞ 9 Strictly speaking, the U 0 and U 1 n... |

145 |
On certain sets of integers
- Roth
- 1953
(Show Context)
Citation Context ...or instance [22] for the standard “colour focusing” proof; another proof can be found in [36]. This theorem was then generalized substantially in 1975 by Szemerédi [39] (building upon earlier work in =-=[33]-=-, [38]), answering a question of Erdős and Turán [8], as follows: Theorem 1.2 (Szemerédi’s theorem). For any integer k ≥ 1 and real number 0 < δ ≤ 1, there exists an integer NSZ(k, δ) ≥ 1 such that fo... |

131 |
der Waerden, Beweis einer Baudetschen Vermutung
- van
- 1927
(Show Context)
Citation Context ...r the use of the Fourier transform or inverse theorems from additive combinatorics. It also gives explicit (but extremely poor) quantitative bounds. 1 Introduction A famous theorem of van der Waerden =-=[44]-=- in 1927 states the following. Theorem 1.1 (Van der Waerden’s theorem). [44] For any integers k, m ≥ 1 there exists an integer N = NvdW(k, m) ≥ 1 such that every colouring c : {1, . . . , N} → {1, . .... |

128 |
Regular partitions of graphs
- Szemerédi
- 1978
(Show Context)
Citation Context ... Gowers, or are instead uniformly almost periodic, or are simply small in L 2 . This theorem is something of a hybrid between the Furstenberg structure theorem [15] and the Szemerédi regularity lemma =-=[40]-=-. A similar structure theorem was a key component to [23]. The fact that the error tolerance in (4) does not go to zero as M → ∞ is crucial in order to obtain this insensitivity to the choice of right... |

112 | A new proof of Szemerédi’s theorem for arithmetic progressions of length 4
- Gowers
- 1998
(Show Context)
Citation Context ...ase k = 3 is by now relatively well understood (see [33], [11], [35], [37], [6], [25], [7] for a variety of proofs). The case k = 4 also has a number of fairly straightforward proofs (see [38], [34], =-=[19]-=-, [9]), although already the arguments here are more sophisticated than for the k = 3 case. However for the case of higher k, only four types of proofs are currently known, all of which are rather dee... |

98 | Hypergraph regularity and the multidimensional Szemerédi theorem - Gowers - 2007 |

91 |
Triple systems with no six points carrying three triangles
- Ruzsa, Szemerédi
- 1976
(Show Context)
Citation Context ...his theorem, however, it is still regarded as a very difficult result, except when k is small. The cases k = 1, 2 are trivial, and the case k = 3 is by now relatively well understood (see [33], [11], =-=[35]-=-, [37], [6], [25], [7] for a variety of proofs). The case k = 4 also has a number of fairly straightforward proofs (see [38], [34], [19], [9]), although already the arguments here are more sophisticat... |

87 | On some sequences of integers
- Erdös, Turán
- 1936
(Show Context)
Citation Context ...roof; another proof can be found in [36]. This theorem was then generalized substantially in 1975 by Szemerédi [39] (building upon earlier work in [33], [38]), answering a question of Erdős and Turán =-=[8]-=-, as follows: Theorem 1.2 (Szemerédi’s theorem). For any integer k ≥ 1 and real number 0 < δ ≤ 1, there exists an integer NSZ(k, δ) ≥ 1 such that for every N ≥ NSZ(k, δ), every set A ⊂ {1, . . . , N} ... |

84 |
Polynomial extensions of van der Waerden’s and Szemerédi’s theorems
- Bergelson, Leibman
- 1996
(Show Context)
Citation Context ...This ergodic theory argument is the shortest and most flexible of all the known proofs, and has been the most successful at leading to further generalizations of Szemerédi’s theorem (see for instance =-=[3]-=-, [5], [12], [13], [14]). On the other hand, the infinitary nature of the argument means that it does not obviously provide any effective bounds for the quantity NSZ(k, δ). The third proof is more rec... |

83 |
Nonconventional ergodic averages and nilmanifolds
- Host, Kra
(Show Context)
Citation Context ...relationship between the two seems very closely related to the distinction in ergodic theory between k − 2-step nilsystems and systems which contain polynomial eigenfunctions of order k −2; see [16], =-=[26]-=- for further discussion of this issue. It is also closely related to the rather vaguely defined issue of distinguishing “almost polynomial” or “almost multilinear” functions from “genuinely polynomial... |

77 | An ergodic Szemerédi theorem for commuting transformation
- Furstenberg, Katznelson
- 1979
(Show Context)
Citation Context ...ic theory argument is the shortest and most flexible of all the known proofs, and has been the most successful at leading to further generalizations of Szemerédi’s theorem (see for instance [3], [5], =-=[12]-=-, [13], [14]). On the other hand, the infinitary nature of the argument means that it does not obviously provide any effective bounds for the quantity NSZ(k, δ). The third proof is more recent, and is... |

69 | The counting lemma for regular k-uniform hypergraphs. Random Structures Algorithms
- Nagle, Rödl, et al.
(Show Context)
Citation Context ...wth), but also requires far more analytic machinery and quantitative estimates. Finally, very recent arguments of Gowers [21] and Rödl, Skokan, Nagle, Tengan, Tokushige, and Schacht [29], [30], [31], =-=[28]-=-, relying primarily on a hypergraph version of the Szemerédi regularity lemma, have been discovered; these arguments are somewhat similar in spirit to Szemerédi’s original proof (as well as the proofs... |

68 | Regularity lemma for k-uniform hypergraphs
- Rödl, Skokan
(Show Context)
Citation Context ...iterated tower growth), but also requires far more analytic machinery and quantitative estimates. Finally, very recent arguments of Gowers [21] and Rödl, Skokan, Nagle, Tengan, Tokushige, and Schacht =-=[29]-=-, [30], [31], [28], relying primarily on a hypergraph version of the Szemerédi regularity lemma, have been discovered; these arguments are somewhat similar in spirit to Szemerédi’s original proof (as ... |

64 |
Lower bounds of tower type for Szemerédi’s uniformity lemma
- Gowers
- 1997
(Show Context)
Citation Context ...ge in order to eventually attain such uniformity. In Szemerédi’s regularity lemma, for instance, one is forced to lose constants which are of tower-exponential type in the regularity parameter ε; see =-=[18]-=-. In the ergodic theory arguments, the situation is even worse; the tower of invariant factors given by Furstenberg’s structure theorem (the ergodic theory analogue of Szemerédi’s regularity lemma) ca... |

60 | Beweis einer Baudetschen Vermutung, Nieuw Arch. Wiskunde 19 - Waerden - 1927 |

56 |
Primitive recursive bounds for van der Waerden numbers
- Shelah
- 1988
(Show Context)
Citation Context ... , N} of cardinality k on which c is constant). the electronic journal of combinatorics 13 (2006), #R99 1sSee for instance [22] for the standard “colour focusing” proof; another proof can be found in =-=[36]-=-. This theorem was then generalized substantially in 1975 by Szemerédi [39] (building upon earlier work in [33], [38]), answering a question of Erdős and Turán [8], as follows: Theorem 1.2 (Szemerédi’... |

56 |
On sets of integers containing no four elements in arithmetic progression
- Szemerédi
- 1969
(Show Context)
Citation Context ...tance [22] for the standard “colour focusing” proof; another proof can be found in [36]. This theorem was then generalized substantially in 1975 by Szemerédi [39] (building upon earlier work in [33], =-=[38]-=-), answering a question of Erdős and Turán [8], as follows: Theorem 1.2 (Szemerédi’s theorem). For any integer k ≥ 1 and real number 0 < δ ≤ 1, there exists an integer NSZ(k, δ) ≥ 1 such that for ever... |

50 |
A density version of the Hales-Jewett theorem for k = 3
- Furstenberg, Katznelson
(Show Context)
Citation Context ...gument is the shortest and most flexible of all the known proofs, and has been the most successful at leading to further generalizations of Szemerédi’s theorem (see for instance [3], [5], [12], [13], =-=[14]-=-). On the other hand, the infinitary nature of the argument means that it does not obviously provide any effective bounds for the quantity NSZ(k, δ). The third proof is more recent, and is due to Gowe... |

49 | Universal Characteristic Factors and FurstenbergAverages
- Ziegler
(Show Context)
Citation Context ...orics 13 (2006), #R99 12s(essentially) proven. The structure theorem seems to correspond to the existence of a universal characteristic factor for Szemerédi-type recurrence properties (see e.g. [26], =-=[47]-=- for a discussion), although unlike the papers in [26], [47] we do not attempt to characterise this factor in terms of nilflows here. The recurrence theorem is very similar in spirit to k−2 iterations... |

48 |
On certain sets of positive density
- Varnavides
- 1959
(Show Context)
Citation Context ...ssion x, x − r, . . . , x − (k − 1)r, as desired. Remark 2.6. One can easily reverse this implication and deduce Theorem 2.4 from Theorem 1.2; the relevant argument was first worked out by Varnavides =-=[45]-=-. In the ergodic theory proofs, Szemerédi’s theorem is also stated in a form similar to (3), but with ZN replaced by an arbitrary measure-preserving system (and r averaged over some interval {1, . . .... |

47 | Integer sets containing no arithmetic progressions
- Heath-Brown
- 1987
(Show Context)
Citation Context ...ver, it is still regarded as a very difficult result, except when k is small. The cases k = 1, 2 are trivial, and the case k = 3 is by now relatively well understood (see [33], [11], [35], [37], [6], =-=[25]-=-, [7] for a variety of proofs). The case k = 4 also has a number of fairly straightforward proofs (see [38], [34], [19], [9]), although already the arguments here are more sophisticated than for the k... |

39 |
Extremal problems on set systems, Random Structures Algorithms 20
- Frankl, Rödl
- 2002
(Show Context)
Citation Context ...= 3 is by now relatively well understood (see [33], [11], [35], [37], [6], [25], [7] for a variety of proofs). The case k = 4 also has a number of fairly straightforward proofs (see [38], [34], [19], =-=[9]-=-), although already the arguments here are more sophisticated than for the k = 3 case. However for the case of higher k, only four types of proofs are currently known, all of which are rather deep. Th... |

34 | Applications of the regularity lemma for uniform hypergraphs. Random Structures Algorithms
- Rödl, Skokan
(Show Context)
Citation Context ...ed tower growth), but also requires far more analytic machinery and quantitative estimates. Finally, very recent arguments of Gowers [21] and Rödl, Skokan, Nagle, Tengan, Tokushige, and Schacht [29], =-=[30]-=-, [31], [28], relying primarily on a hypergraph version of the Szemerédi regularity lemma, have been discovered; these arguments are somewhat similar in spirit to Szemerédi’s original proof (as well a... |

29 | The ergodic theoretical proof of Szemerédi’s theorem
- Furstenberg, Katznelson, et al.
- 1982
(Show Context)
Citation Context ...pe), due mainly to the reliance on the van der Waerden theorem and the regularity lemma, both of which have notoriously bad dependence of the constants. Shortly afterwards, Furstenberg [10] (see also =-=[15]-=-, [11]) introduced what appeared to be a completely different argument, transferring the problem into one of recurrence in ergodic theory, and solving that problem by a number of ergodic theory techni... |

25 |
An ergodic Szemerédi theorem for IP-systems and combinatorial theory
- Furstenberg, Katznelson
- 1985
(Show Context)
Citation Context ...ory argument is the shortest and most flexible of all the known proofs, and has been the most successful at leading to further generalizations of Szemerédi’s theorem (see for instance [3], [5], [12], =-=[13]-=-, [14]). On the other hand, the infinitary nature of the argument means that it does not obviously provide any effective bounds for the quantity NSZ(k, δ). The third proof is more recent, and is due t... |

20 |
A note on a question of Erdos and
- Solymosi
(Show Context)
Citation Context ...eorem, however, it is still regarded as a very difficult result, except when k is small. The cases k = 1, 2 are trivial, and the case k = 3 is by now relatively well understood (see [33], [11], [35], =-=[37]-=-, [6], [25], [7] for a variety of proofs). The case k = 4 also has a number of fairly straightforward proofs (see [38], [34], [19], [9]), although already the arguments here are more sophisticated tha... |

20 | The Gaussian primes contain arbitrarily shaped constellations
- Tao
(Show Context)
Citation Context ... recent hypergraph approach of Gowers [21] and of Rödl-Skokan-Nagle-Schacht [28], [29], [30] seems to have a decent chance of being “relativized” to pseudorandom sets such as the “almost primes”; see =-=[43]-=-). Indeed, some of the work used to develop this paper became incorporated into [23], and conversely some of the progress developed in [23] was needed to conclude this paper. Remark 1.4. It is certain... |

15 | Set-polynomials and polynomial extension of HalesJewett theorem
- Bergelson, Leibman
- 1999
(Show Context)
Citation Context ...ergodic theory argument is the shortest and most flexible of all the known proofs, and has been the most successful at leading to further generalizations of Szemerédi’s theorem (see for instance [3], =-=[5]-=-, [12], [13], [14]). On the other hand, the infinitary nature of the argument means that it does not obviously provide any effective bounds for the quantity NSZ(k, δ). The third proof is more recent, ... |

13 | Combinatorial proofs of the polynomial van der Waerden theorem and the polynomial Hales-Jewett theorem
- Walters
(Show Context)
Citation Context ...f this lemma in terms of conditional expectation; see [41]. the electronic journal of combinatorics 13 (2006), #R99 5sgeneral result, but fortunately such generalizations are known to exist (see e.g. =-=[46]-=- for further discussion). In principle, the quantitative ergodic approach could in fact have a greater reach than the traditional ergodic approach to these problems; for instance, the recent establish... |

12 | A mean ergodic theorem for (1/N) ∑N n=1 f(T nx)g(T n2x). Convergence in ergodic theory and probability 92 - Weiss - 1993 |

11 |
Irregularities of sequences relative to arithmetic progressions, IV.Period.Math
- Roth
- 1972
(Show Context)
Citation Context ...t of the more well-known density incrementation argument which appears in several proofs of Szemerédi’s theorem (starting with Roth’s original argument [33], but see also [19], [20], [25], [7], [38], =-=[34]-=-, [39]). In that strategy one passes from the original set {1, . . . , N} to a decreasing sequence of similarly structured subsets (e.g. arithmetic progressions or Bohr sets) while forcing the density... |

10 |
Selected topics from the metric theory of dynamical systems
- Rohlin
- 1949
(Show Context)
Citation Context ...ly the act of averaging a function on each atom 3 . As such, we do not need such results from measure theory as the construction of product measure (or conditional product measure, via Rohlin’s lemma =-=[32]-=-), which plays an important part of the ergodic theory proof, notably in obtaining the structure and recurrence theorems. Also, we do not use the compactness of Hilbert-Schmidt or Volterra integral op... |

9 |
An inverse theorem for the Gowers U 3 (G) norm, preprint
- Green, Tao
(Show Context)
Citation Context ... admit quadratic eigenfunctions; see e.g. [16] for further discussion. Intriguingly, hints of this “generalized quadratic” structure also emerge in the work of Gowers [19]. For further discussion see =-=[24]-=-. The structure theorem, Theorem 3.5, can be viewed as some sort of duality relationship between UAP k−2 and U k−1 . We now provide two demonstrations of this duality. The first such demonstration is ... |

8 | Pointwise convergence of ergodic averages along cubes
- Assani
(Show Context)
Citation Context ...P k−2 with �F �UAP k−2 ≤ 1 such that |〈f, F 〉| ≥ ε2k−1. Proof. We need the concept of a dual function from [23]; the ergodic theory analogue of such functions have also been recently studied in [26], =-=[1]-=-. For any function f : ZN → C and any d ≥ 0, we define the dual function of order d of f, denoted Dd(f), by the recursive formula D0(f) := 1 (18) (i.e. D0(f) is just the constant function 1) and for a... |

6 |
The complexity of the collection of measure-distal transformations. Ergodic Theory Dynam
- Beleznay, Foreman
- 1996
(Show Context)
Citation Context ...orse; the tower of invariant factors given by Furstenberg’s structure theorem (the ergodic theory analogue of Szemerédi’s regularity lemma) can be as tall as any countable ordinal, but no taller; see =-=[2]-=-. Remark 4.7. In the ergodic theory setting, one can also define analogues of the U k−1 norms, giving rise to the concept of invariant factors whose complement consists entirely of functions which are... |

6 | Aspects of uniformity in recurrence - Bergelson, Host, et al. |

6 |
On triples in arithmetic progression, GAFA 9
- Bourgain
- 1999
(Show Context)
Citation Context ...t is still regarded as a very difficult result, except when k is small. The cases k = 1, 2 are trivial, and the case k = 3 is by now relatively well understood (see [33], [11], [35], [37], [6], [25], =-=[7]-=- for a variety of proofs). The case k = 4 also has a number of fairly straightforward proofs (see [38], [34], [19], [9]), although already the arguments here are more sophisticated than for the k = 3 ... |

6 | Density theorems and extremal hypergraph problems, manuscript
- Rödl, Schacht, et al.
(Show Context)
Citation Context ...er growth), but also requires far more analytic machinery and quantitative estimates. Finally, very recent arguments of Gowers [21] and Rödl, Skokan, Nagle, Tengan, Tokushige, and Schacht [29], [30], =-=[31]-=-, [28], relying primarily on a hypergraph version of the Szemerédi regularity lemma, have been discovered; these arguments are somewhat similar in spirit to Szemerédi’s original proof (as well as the ... |

6 |
Szemeredi’s regularity lemma revisited, preprint
- Tao
- 2005
(Show Context)
Citation Context ...more 3 Readers familiar with the Szemerédi regularity lemma may see parallels here with the proof of that lemma. Indeed one can phrase the proof of this lemma in terms of conditional expectation; see =-=[41]-=-. the electronic journal of combinatorics 13 (2006), #R99 5sgeneral result, but fortunately such generalizations are known to exist (see e.g. [46] for further discussion). In principle, the quantitati... |

6 | Note on a generalization of Roth’s theorem - Solymosi - 2003 |

5 |
A Szemeredi-type theorem for sets of positive density
- Bourgain
(Show Context)
Citation Context ... however, it is still regarded as a very difficult result, except when k is small. The cases k = 1, 2 are trivial, and the case k = 3 is by now relatively well understood (see [33], [11], [35], [37], =-=[6]-=-, [25], [7] for a variety of proofs). The case k = 4 also has a number of fairly straightforward proofs (see [38], [34], [19], [9]), although already the arguments here are more sophisticated than for... |

3 |
A mean ergodic theorem for 1/N ∑N n=1 f(T n x)g(T n2 x), in Convergence in ergodic theory and probability
- Furstenberg, Weiss
- 1993
(Show Context)
Citation Context ...r resemblance with any quadratic phase function. These “generalized quadratic phase functions” are related to 2-step nilsystems, which are known to not always admit quadratic eigenfunctions; see e.g. =-=[16]-=- for further discussion. Intriguingly, hints of this “generalized quadratic” structure also emerge in the work of Gowers [19]. For further discussion see [24]. The structure theorem, Theorem 3.5, can ... |

2 |
Herbrand’s theorem and proof theory
- Girard
- 1982
(Show Context)
Citation Context ...f 1 It may also be possible in principle to extract some bound for NSZ(k, δ) directly from the original Furstenberg argument via proof theory, using such tools as Herbrand’s theorem; see for instance =-=[17]-=- where a similar idea is applied to the Furstenberg-Weiss proof of van der Waerden’s theorem to extract Ackermann-type bounds from what is apparently a nonquantitative argument. However, to the author... |

1 | Irregularities of sequences relative to arithemtic progressions - Roth - 1972 |

1 | An information-theoretic proof of Szemerédi’s regularity lemma, unpublished - Tao |